Simple Harmonic Motion Calculator | SHM Physics Calculator Online

Input Parameters

Amplitude (A) - Maximum displacement

Angular Frequency (ω) - rad/s

OR Frequency (f) - Hz (cycles/second)

OR Period (T) - seconds

Time (t) - seconds (for position calculation)

Initial Phase (φ) - radians

Mass (m) - kg (for energy calculations)

📊 Calculation Results

Angular Frequency (ω) -- rad/s

Frequency (f) -- Hz

Period (T) -- s

Position at time t (x) -- m

Velocity at time t (v) -- m/s

Acceleration at time t (a) -- m/s²

Maximum Velocity (v_max) -- m/s

Maximum Acceleration (a_max) -- m/s²

Total Energy (E) -- J

📐 SHM Formulas Used

x(t) = A × cos(ωt + φ)

v(t) = -Aω × sin(ωt + φ)

a(t) = -Aω² × cos(ωt + φ)

ω = 2πf = 2π/T

E = ½mω²A²

🎬 SHM Animation

Understanding Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. This simple harmonic motion calculator helps you analyze all aspects of SHM.

Key Characteristics of SHM

Amplitude (A): Maximum displacement from the equilibrium position, measured in meters
Period (T): Time taken for one complete oscillation, measured in seconds
Frequency (f): Number of oscillations per second, measured in Hertz (Hz)
Angular Frequency (ω): Rate of change of phase angle, measured in radians per second
Phase (φ): Initial angle that determines the starting position

How to Use This Simple Harmonic Motion Calculator

Enter the amplitude of oscillation
Provide either angular frequency, frequency, or period (the calculator will derive the others)
Optionally enter a specific time to calculate position, velocity, and acceleration at that instant
Add initial phase if the motion doesn't start from maximum displacement
Include mass to calculate total mechanical energy
Click "Calculate SHM" to see all results

Real-World Examples of SHM

Pendulum clocks (for small angles)
Mass-spring systems
Vibrating guitar strings
Molecular vibrations
LC electrical circuits

Frequently Asked Questions

What is simple harmonic motion?

Simple harmonic motion (SHM) is oscillatory motion where the restoring force is proportional to displacement. Examples include a mass on a spring and a simple pendulum (for small angles). The motion follows a sinusoidal pattern.

How do I calculate the period of SHM?

The period T can be calculated using T = 2π/ω or T = 1/f, where ω is angular frequency and f is frequency. For a mass-spring system, T = 2π√(m/k), and for a simple pendulum, T = 2π√(L/g).

What is the relationship between frequency and period?

Frequency and period are inversely related: f = 1/T. If an object completes one oscillation in 2 seconds (T = 2s), its frequency is 0.5 Hz.

When is velocity maximum in SHM?

Velocity is maximum when the object passes through the equilibrium position (x = 0). At this point, v_max = Aω, where A is amplitude and ω is angular frequency.

When is acceleration maximum in SHM?

Acceleration is maximum at the extreme positions (x = ±A) where the restoring force is greatest. The maximum acceleration is a_max = Aω².

Is this simple harmonic motion calculator free?

Yes! This SHM calculator is completely free to use with no limitations. It's designed for students, teachers, and physics enthusiasts.